How to make mesh lights work in Blender Eevee?
Mesh lights can be used with EEVEE but they require an extra step to work: baking indirect light. 


Here is a scene with a Light-type object (correctly lighting a receiver plane) and two Emission meshes (currently doing nothing)


We should add an Irradiance Volume object, and make it encompass the volume that is affected by the light that we want to capture. This object has a Resolution option that should be adjusted depending on needs: the finer the grid, the most spatially accurate the description of the light in the scene.

  
To finish, in the Scene properties, under Indirect Lighting, we must hit "Bake Indirect Lighting". This will compute how much diffuse light is bouncing around each one of the "dots" of the Irradiance grid.



Now the mesh lights are contributing to the lighting in the scene! Moreover, once the lighting is baked (and if the lights are not moved), objects can still be moved around and will be correctly lit by the baked light (they won't cast shadows though!):




As far as I know, the Indirect Light cache doesn't have frames so the animation of mesh lights isn't possible at the moment.



In the manual: Irradiance Volumes - Indirect Lighting Cache